ODTK A Technical Summary

Transcription

1 ODTK A Technical Summary Analytical Graphics, Inc. February 17, 2009 Contents 1 Introduction Optional States Optional Central Bodies Optional Force Models Optional Measurement Models Measurement Model Options I Orbit Determination Science 7 2 Methods of Orbit Determination Initial Orbit Determination Least Squares Di erential Corrections Sequential Processing ODTK Optimal Orbit Determination Mathematical Operators for SP Methods Subscript Notation Nonlinear Operators Linear Operators De nitions State Estimate Reference for Linearization of SP Methods Local Linearization Global Linearization Observability Completeness Optimal Orbit Determination Discussion Sherman s Theorem Complete State Estimate Local Linearization c Analytical Graphics, Inc

2 4 Measurements Ground Station Space Based TDRSS GPS Optimal Sequential Filter State Estimate Error Model Integral Equation State Error Covariance for Filter Time Update Solutions for the Optimal State Estimate Error Model Gravity Solution Air-Drag Solution Solar Pressure Solution Filter Time Update Algorithm Filter Measurement Update Algorithm Filter Measurement Update and Filter Time Update Measurement Editing Kalman Measurement Editor Limitation Filter Initialization Problem The Dynamic Editor Filter Initialization Filter Divergence Kalman Measurement Editor Dynamic Editor Unmodeled Thrust Maneuvers Fixed Interval Sequential Smoother (FIS) FIS Initialization Notation for Smoother Nonlinear State Transition FIS Sequential Equations Transition Smoothed State Estimate Backwards Incorporate Filter Estimate and Covariance at Time t k Prepare to Calculate Smoother Covariance Smoother Covariance Filter-Smoother Consistency Test Criterion Variable-Lag Smoother 18 8 Carlton-Rauch Fixed-Epoch Smoother (CR-FES) FES Initialization Measurement at t j = t k Filter CR-FES Filter Measurements at t j = t k+1 ; t k+2 ; : : : Frazer Fixed-Epoch Smoother (F-FES) FES Initialization Measurement at t j = t k Filter F-FES Filter Measurements at t j = t k+1 ; t k+2 ; : : :

3 10 Least Squares 21 II Computer Science Software Architecture AGI Technology Platform (ATP) ODTK Engine ODTK GUI Summary 25 1 Introduction ODTK is an optimal orbit determination capability, where optimality is de ned in the technical discussion in Section 3 below. ODTK is an integrated set of software modules, using the same astrodynamics software library and integrators as STK, using modern C++ programming techniques, designed to support realtime satellite operations, non-real-time post-mission analysis, and design and trade studies. The software architecture is described in Part II, below. The main modules in ODTK are a Filter, a Smoother, and a Simulator. The Filter is an optimal sequential lter, as described below. The Filter is a particular nonlinear extension to the linear Kalman Filter[4], and is signi cantly di erent in the way that process noise is set by the user and implemented in the Time Update. The ODTK approach improves stability, improves covariance realism, and provides the user with physically intuitive process noise controls. The ODTK Smoother is a nonlinear extension to the Meditch[5] linear xed-interval smoother. The Simulator can deviate initial orbit conditions, force model coe cients, facility and transponder biases, clock behavior, and tracking facility locations, all under analyst controls. 1.1 Optional States The following list of estimable states re ects those quantities that ODTK can deviate during simulation and estimate during lter and smoother operations. Any number of states from any number of satellites and any number of trackers can be simulated and/or estimated simultaneously with ODTK. 3

4 Satellite Position and velocity Drag coe cient Local atmospheric density Solar radiation pressure coe cient(s) One coe cient for most satellites Two coe cients for GPS Satellites Measurement biases Time-varying time-correlated bias on each measurement type Time-varying time-correlated bias for each transponder including Relay satellite transponder SGLS-type transponder TDRS-type and BRTS-type transponder TDOA / FDOA-type transponder GEO transponder for GPS signal Facility location Troposphere scale correction Space based GPS receiver Clock phase, frequency, and (optional) aging Antenna location in body frame Center-of-mass location in body frame GPS Satellite transmitter Clock phase, frequency and (optional) aging Finite maneuver magnitude and/or direction 1.2 Optional Central Bodies ODTK can perform simulation, lter and smoother operations relative to the Earth, Moon or Sun. 1.3 Optional Force Models The following force models are provided, each implementation conforms to international standards, as de ned in IERS Conventions and comparable sources. Gravitational perturbations Numerous Earth gravity- eld models including: EGM2008, EGM96, WGS84 JGM2, JGM3 GEM-series models GGM01C, GGM02C Several Moon gravity- eld models for Moon-orbiting objects, which the user can supplement Luni-solar perturbations 4

5 Planetary perturbations (using the barycenter of respective planet moon systems) Solid-earth tide perturbations Ocean tide perturbations Relativistic accelerations Atmospheric drag perturbations Spherical body model Plug-in point for user-provided complex force model Several atmospheric density models including CIRA 1972, Jacchia 1971 Jacchia-Roberts MSIS 1990 NRLMSIS 2000 Photon pressure perturbations Spherical body model for solar photon pressure Spherical body model for earth albedo pressure GPS-speci c block-speci c models for GPS satellites Plug-in point for user-provided complex force model Eclipse model for earth shadow and lunar shadow User-selectable le for Earth albedo-model coe cients Thrust perturbations Impulsive delta-v Finite maneuvers (constant thrust or complex acceleration pro les) Grouped thrust events to re ect repeated use of the same thrusters 1.4 Optional Measurement Models The following measurement types represent capabilities of ODTK. A measurement provider plug-in point is provided which will accommodate any measurement format. All measurement types that can be ltered can also be simulated. Ground-based tracking Range, azimuth, elevation, Doppler Right ascension and declination Satellite laser ranging X, Y angles Direction Cosines Deep Space Network (DSN) Doppler, three-way Doppler, TCP, three-way TCP, sequential ranging Multiply relayed ground tracking Bistatic ranging Time-Di erence of Arrival (TDOA), Frequency-Di erence of Arrival (FDOA), TDOA rate, singly di erenced TDOA & FDOA 5

6 TDRS 1 4-legged round-trip range, 5-legged Doppler, Return-link Doppler BRTS 2 range and Doppler GPS 2-legged pseudo-range (GPS to GEO to ground) GPS receiver tracking of GPS signal C/A pseudo-range & L/A phase (also called ADR) L1 & L1 Phase P1 & P2 pseudo-range Dual frequency corrected pseudo-range and/or phase Single-di erenced pseudo-range and phase Double-di erenced pseudo-range and phase Space based tracking Range, azimuth, and elevation Right ascension & declination Space based TDOA, TDOA rate, & FDOA 1.5 Measurement Model Options Various high delity models are employed to support high accuracy programs. Earth-based station motion models Solid earth tides Polar tide Ocean loading Tectonic plate drift Antenna-correction models (DSN receivers) Ground based physical models Ionospheric refraction (IRI) model Tropospheric refraction models SCF model Marini-Murray (for laser ranging) Saastamoinen with Niell hydrostatic mapping function Satellite body motion models Antenna location de ned in body coordinates Body orientation de ned by Attitude rules Attitude pro le (quaternions) Attitude model (GPS satellites) 1 TDRS = Tracking and Data Relay Satellite 2 Bilateral Ranging Transponder System 6

7 Part I Orbit Determination Science 2 Methods of Orbit Determination Orbit determination methods are sharply partitioned by three classes: Initial Orbit Determination (IOD), Least Squares (LS), and Sequential Processing (SP). Operationally, the order in which these methods are used de nes a dependency tree: IOD output is LS input, and LS output is SP input: 2.1 Initial Orbit Determination IOD =) LS =) SP IOD methods input tracking measurements with tracking platform locations, and output spacecraft position and velocity estimates. No a priori orbit estimate is required. Associated output orbit estimation error magnitudes are large. IOD methods are nonlinear methods, and are relatively trivial to implement. Measurement editing is typically not performed during IOD calculations. Operationally, the orbit determination process is frequently begun, or restarted, with IOD. IOD methods were derived by various authors: LaPlace, Poincaré, Gauss, Lagrange, Lambert, Gibbs, Herrick, Williams, Stumpp, Lancaster, Blanchard, Gooding, and Smith. ODTK provides two methods to solve initial orbit determination problems: The Herrick-Gibbs method for range and angles measurements, and the Gooding method for angles-only. 2.2 Least Squares Di erential Corrections LS methods input tracking measurements with tracking platform locations and a prior orbit estimate, and output a re ned orbit estimate. A prior orbit estimate is required. Associated output error magnitudes are small when compared to IOD outputs. LS methods consist of a sequence of linear LS corrections where sequence convergence is de ned as a function of tracking measurement residual RMS (root mean square). Each linear LS correction is characterized by a minimization of the sum of squares of tracking measurement residuals. LS methods produce re ned orbit estimates in a batch mode, together with error covariance matrices that are optimistic; i.e., orbit element error variances are typically too small by at least an order of magnitude. Operationally, LS may be the only method used, or it may be used to initialize SP. LS methods frequently require inspection and manual measurement editing by human intervention. LS algorithms therefore require elaborate software mechanisms for measurement editing. The LS method was derived rst by Gauss[1] in 1795, and then independently by Legendre. ODTK provides a QR factorization and triangularization method with orthogonal Householder transformations to solve the LS equation. 2.3 Sequential Processing SP methods input tracking measurements with tracking platform locations, input an a priori state estimate (inclusive of orbit estimate), and input an a priori state error covariance matrix. A prior state estimate is required, and a prior state error covariance matrix is required. SP methods output re ned state estimates in a sequential mode. SP lter methods are forward-time recursive sequential machines consisting of a repeating pattern of lter time update of the state estimate and lter measurement update of the state estimate. The lter time update propagates the state estimate forward, and the lter measurement update incorporates the next measurement. The recursive pattern includes an important interval of lter initialization. SP smoother methods are backward-time recursive sequential machines consisting of a repeating pattern of state estimate re nement using lter outputs and backwards transition. Time transitions for both lter and smoother are dominated most signi cantly by numerical orbit propagators. The search for sequential processing was begun by Wiener, Kalman[4], Bucy[7], and others. ODTK provides a unique sequential lter-smoother to solve the optimal orbit determination problem. 7

8 2.4 ODTK ODTK provides methods for IOD, LS, and SP. In particular, IOD and LS are used to initialize SP. 3 Optimal Orbit Determination Orbit determination refers to the estimation of orbits of spacecraft (or natural satellites or binary stars) relative to primary celestial bodies, given applicable measurements. All useful orbit determination methods produce orbit estimates, and all orbit estimates have errors. But what is optimal orbit determination? By itself, the adjective optimal refers [44] to most desirable, most favorable, or most satisfactory. But most satisfactory to whom? There are choices to make from available orbit determination methods. The fastest methods are the least accurate. Should we prefer sequential methods to batch methods? Should we minimize the size of measurement residuals or the size of orbit errors? How should we model measurement residuals and orbit errors? All orbit determination problems are multidimensional and nonlinear. Should we attempt a multidimensional nonlinear solution directly? Or should we use a linearization method? If so, is there a preferred method for linearization? The purpose of this section is to answer the question: What is optimal orbit determination? A de nition requires the use of mathematical operators for sequential processing (SP) methods. 3.1 Mathematical Operators for SP Methods Subscript Notation State Matrices The state estimate ^X is referred to two times with the notation[5]: ^X jji ^X (t j jt i ) (1) where i, j 2 f0; 1; 2; : : :g. The time t j to the left of the vertical bar denotes the epoch for ^X, and is driven by the lter time update function. The time t i to the right of the bar denotes the time-tag of the last measurement processed to form ^X, and is driven by the lter measurement update function. Examples: ^X 7j6 refers to the state estimate at time t 7, given the last measurement processed at time t 6, whereas ^X 7j7 refers to the state estimate at time t 7, given the last measurement processed at time t 7. Evidently, ^X7j6 was obtained by lter time update of ^X6j6 from t 6 to t 7. Similar notation is used for the state estimate correction: and the state estimate error covariance matrix: ^X jji ^X (t j jt i ) (2) P jji P (t j jt i ) (3) Measurement Matrices Denote a measurement at time t j with y j, and denote a measurement estimate (representation) at time t j with ^y jjh. The subscript h denotes the time t h of last measurement incorporated by the lter Nonlinear Operators Nonlinear operators are required in the state estimate time update and the state estimate measurement update for SP methods of orbit determination. State Propagation t i to time t j : Let ' denote a nonlinear operator that propagates the state estimate ^X ijh from time ^X jjh = ' nt j ; ^X o ijh ; t i (4) 8

9 Measurement Representation Let y () denote a nonlinear operator that calculates the measurement representation ^y jjh, given the state estimate ^X jjh : ^y jjh = y ^Xjjh ; t j (5) Linear Operators State Estimate Error Propagation Let j;i (t j ; t i ) denote the linear operator that propagates the state error estimate ^X ijh from time t i to time t j where ^X jjh = j;i ^X ijh j;i i ^X jjh (7) and where evaluation derives from ^X jjh. Measurement Residual time t j : Let y j denote the linear operator that de nes the measurement residual at y j = y j ^y jjh (8) Measurement/State Partials Jacobian derivatives at time t j : Let H j denote the jacobian of measurement to state H j = j ^X jjh where evaluation derives from ^X jjh. 3.2 De nitions State Estimate Reference for Linearization of SP Methods Evaluation of the measurement representation ^y j de ned by Eq. 5 requires the use of some a priori state estimate reference ^X jjh, where t j t h. A similar requirement is associated with Eqs. 7 and 9. ^Xjjh is called the state estimate reference for linearization Local Linearization Let t k and t k+1 t k be the time tags of adjacent ordered measurements y k and y k+1, for k 2 f0; 1; 2; : : :g. That is, there exist no measurements between y k and y k+1. Then the use of ^Xk+1jk as the state estimate reference for all linearizations at time t k+1 de nes local linearization at time t k+1. The use of any state estimate reference other than ^X k+1jk at time t k+1 for linearization is a non-local linearization at time t k Global Linearization Given the integer variable k 2 f0; 1; 2; : : :g and any xed non-negative integer j, then the use of ^Xkjj as the state estimate reference for linearization at time t k for each k is global linearization Observability A particular parameter is observable to a particular measurement if and only if the sequential processing of that measurement reduces the estimate error variance on that parameter. 9

10 3.2.5 Completeness The state estimate structure is complete if and only if all parameters, that are both unknown and observable, are contained in the state estimate structure Optimal Orbit Determination By optimal orbit determination [24], we mean that the method used to calculate the state estimate (containing the orbit estimate) satis es the following eight conditions: 1. Sequential processing (SP) is used to account for force modeling errors and measurement information in the time order in which they are realized. 2. Sherman s Theorem (summary) [2],[3],[5],[4]: The optimal state estimate correction matrix ^X is the expectation of the state error matrix X given the measurement residual matrix y. That is: ^X = E fxjyg. 3. Linearizations of state estimate time transition and state to measurement representations are local in time, not global. 4. The state estimate structure is complete. 5. All state estimate models and state estimate error model approximations are derived from appropriate force modeling physics, and measurement sensor performance. 6. All measurement models and measurement error model approximations are derived from appropriate sensor hardware de nitions and associated physics, and measurement sensor performance. 7. Necessary conditions for real data: Measurement residuals approximate Gaussian white noise [5],[8] McReynold s [29],[24] lter-smoother consistency test is satis ed with probability Su cient conditions for simulated data: The state estimate errors agree with the state estimate error covariance function. The rst six conditions de ne standards for optimal algorithm design, and the creation of a realistic state estimate error covariance function. The last two conditions enable validation: They de ne realizable test criteria for optimality. 3.3 Discussion Sherman s Theorem Sherman s Theorem [5] is applicable to a linear state estimate; i.e., a condition where the state estimate is a linear combination of available measurements. But in all orbit determination problems, the orbit (substate) estimate is a nonlinear function of available measurements. We must linearize in order to use Sherman s Theorem. Optimal orbit determination requires measurement linearization about the local state estimate ^X k+1jk to produce a local measurement residual: y k+1 = y k+1 y ^Xk+1jk and linearization about the same local state estimate ^X k+1jk to produce a local state error estimate ^X k+1jk+1, given y k+1. Local linearization enables a linear relation between each state error estimate ^X k+1jk+1 and each measurement residual y k+1. With local linearization, one applies Sherman s Theorem anew to each scalar measurement residual, never simultaneously to a batch of measurement residuals. 10

11 Let X k+1jk = X k+1jk ^Xk+1jk de ne the error in state estimate ^X k+1jk, and let X k+1jk = X k+1jk ^X k+1jk de ne the error in ^X k+1jk. Ideally X k+1jk = 0, and the state error estimate ^X k+1jk is perfect. When X k+1jk 6= 0, assign a penalty (or loss) function L = L X k+1jk with four admissibility requirements: L is a scalar-valued function of the N state estimate variables. L (0) = 0, where the rst 0 is an N 1 matrix of zeros. No loss is assigned when the state error estimate is perfect. L Xk+1jk a L Xk+1jk b whenever Xk+1jk a Xk+1jk b, where is a scalar-valued, nonnegative, convex function of N variables. Thus L is de ned to be a non-decreasing function of distance from the origin. The closer X k+1jk is to zero, the smaller the loss. L X k+1jk = L Xk+1jk. That is, L () is symmetric about the origin. Performance J X k+1jk is de ned as the expectation of the loss; i.e., as the mean value of loss. Our goal is to minimize J X k+1jk : J X k+1jk = E L Xk+1jk Denote the conditional probability distribution function of X k+1jk given y k+1 by: (10) P X k+1jk jy k+1 = F fjy k+1 g (11) The reader is referred to Chapter 5.0, Section 5.2, of Meditch [5] for the following theorems. Most General Form Given any admissible loss function L X k+1jk, and any conditional probability distribution function F fjy k+1 g such that F fjy k+1 g is: Symmetric about its mean Convex for all then: ^X k+1jk+1 = E X k+1jk jy k+1 (12) Application of the conditional mean E X k+1jk jy k+1 generates a global minimum to the performance function J X k+1jk. This is true for all combinations of admissible loss functions and symmetric and convex conditional probability distribution functions. Proof is due to Sherman[2][3]. Gaussian Distribution Given any admissible loss function L X k+1jk, and Gaussian random variables X k+1jk and y k+1, then: ^X k+1jk+1 = E X k+1jk jy k+1 (13) Application of the conditional mean E X k+1jk jy k+1 generates a global minimum to the performance function J X k+1jk, even for asymmetric loss functions. Proof is due to Doob[14]. T Mean Square Error If L X k+1jk = Xk+1jk Xk+1jk, then: ^X k+1jk+1 = E X k+1jk jy k+1 (14) T The loss function X k+1jk n Xk+1jk is referred to as the mean square state error. Minimization of T o the performance function E X k+1jk Xk+1jk results, in part, in the minimization of mean square orbit error. Application of the conditional mean E X k+1jk jy k+1 generates a global minimum to the performance function. In this case the conditional probability distribution function need not be either symmetric or convex. Proof is due to Doob[14]. 11

12 3.3.2 Complete State Estimate Consider any case where the state estimate structure is incomplete. The observable parameter neglected in the state estimate structure will alias into the estimated orbit elements, signi cantly degrading them. Thus one needs an appropriate place in the state estimate structure to put every observable e ect Local Linearization Local linearization has obvious bene ts relating to the use of small local a priori estimation error magnitudes. Another bene t becomes obvious when one considers gravity resonance e ects. Local linearization immediately transfers locally acquired orbit measurement information to the local state estimate for numerical integration by a rigorous nonlinear complete formulation of the equations of motion across all time intervals. Thus the new information relating to all signi cant resonance e ects is rigorously accounted for by numerical integration of the local orbit estimate. Global linearization puts the burden of modeling resonance e ects, in part, into the stochastic process for the gravity error model. The construction of a global stochastic model for gravity errors referred to resonance is daunting no one has even attempted it. 4 Measurements 4.1 Ground Station Two-way range Two-way Doppler (carrier phase count) Azimuth and elevation angles Right ascension and declination angles 4.2 Space Based Right ascension and declination angles Two-way range 4.3 TDRSS Two-way range with four legs Two-way Doppler with ve legs One-way Doppler return link with three legs Two-way BRTS with four legs 4.4 GPS Single frequency C/A code pseudo-range Two frequency P-code pseudo-range Two frequency ionosphere removal USER clock e ects removal by rst di erences Two frequency Doppler carrier phase count measurements 12

13 5 Optimal Sequential Filter 5.1 State Estimate Error Model The optimal model equation for the state error is de ned by the linear stochastic di erential equation: d X (t) = A (t) X (t) + B (t) u (t) (15) dt where X (t) is an n x 1 matrix, where u (t) is a 3 x 1 matrix-valued correlated Gaussian random error, where A (t) is an n x n time dependent matrix, and where B (t) is an n x 3 time dependent matrix. For sequential orbit determination, u (t) is always serially correlated (non-white) due to correlation in modeling errors for gravity[22] [23], air-drag, and solar pressure. 5.2 Integral Equation Eq. 15 has an integral: Z tk+1 X (t k+1 ) = (t k+1 ; t k ) X (t k ) + (t k+1 ; ) B () u () d (16) t k The right-hand side of this equation presents the sum of two nx1 matrix terms. The rst term propagates initial condition state estimate errors X (t k ), and the second term accumulates and propagates acceleration modeling errors u (t) (state error process noise). 5.3 State Error Covariance for Filter Time Update Form the outer product on X t k+1jk using Eq. 16, and take its expectation to de ne: o T P k+1jk = E nx t k+1jk X tk+1jk (17) where: o T o T E nx t k+1jk X tk+1jk = (t k+1 ; t k ) E nx t kjk X tkjk (t k+1 ; t k ) T + and where: I C k+1;k + I L k+1;k + I R k+1;k (18) Z Z tk+1 Ik+1;k C = H (t k+1 ; ) E u () u T (t) H T (t k+1 ; t) ddt t k (19) Z tk+1 Ik+1;k L = (t k+1 ; t k ) t k E X (t k+1 jt) u T (t) H T (t k+1 ; t) dt (20) I R k+1;k = Z tk+1 t k H (t k+1 ; ) E u () X T (t k+1 j) d T (t k+1 ; t k ) (21) Then: where: H (t; ) = (t; ) B () (22) P k+1jk = (t k+1 ; t k ) P kjk (t k+1 ; t k ) T R R + Pk+1;k (23) o T P kjk = E nx t kjk X tkjk (24) and: 13

14 R R Pk+1;k = IC k+1;k + Ik+1;k L + Ik+1;k R (25) Eq. 23 formally speci es the method for moving the optimal state error covariance P kjk from time t k to time t k+1, and for the accumulation of acceleration modeling errors, to get P k+1jk. Examine Eq. 23 to see that optimal state error covariance propagation provides the structure Pk+1;k to accomodate random force modeling errors, whereas the least squares model has no such structure. This term is referred to as process noise covariance. R R Notice from Eq. 23 that the optimal lter covariance time update is time sequential. When P is signi cant and measurements are sparce with time, optimal k+1;k estimation requires a time sequential update so as to incorporate Pk+1;k between measurement times t k and t k+1. Simultaneous batch processing of measurements eliminates the sequential update and thereby destroys optimality. If u were white noise, then the process noise covariance would reduce to Kalman s Q k+1;k matrix. From the de nition of optimality: All state estimate modeling and modeling errors R R are derived from appropriate force modeling physics and sensor performance. That is, the contents of Pk+1;k are not arbitrary. We must seek appropriate results from force modeling physics. 5.4 Solutions for the Optimal State Estimate Error Model Force modeling errors u (t) are captured via the state estimate error covariance function. Most signi cantly they derive from errors on the gravity model, the air-drag model, the solar pressure model, and the spacecraft rocket engine thrust model Gravity Solution A solution to stochastic gravity modeling errors is given by Wright [22] [23]. This solution is derived by an extension to Kaula s gravity error covariance model [21] Air-Drag Solution A new solution to stochastic air-drag modeling errors is presented in the AGI math speci cation [25] for ODTK. In this innovation we sequentially estimate corrections to atmospheric density, with an autonomous lter gain that is de ned (in part) by historical F 10 and K P data collected across two solar cycles, and by near real-time values of F 10 and K P Solar Pressure Solution Stochastic solar pressure error modeling is presented in the AGI math speci cation [25] for ODTK. A geometric two-cone model is used for umbra-penumbra de nition [19], with Earth disc diameter modi cation to compensate for atmospheric refraction e ects [38]. In the absence of spacecraft speci c surface and attitude modeling, the coe cient of di use re ection for a spherical surface is modeled [34]. 5.5 Filter Time Update Algorithm Let t k be the time of last measurement. Given the state estimate ^X kjk, state estimate error covariance matrix P kjk, and a new scalar measurement y k+1 at time t k+1 t k, calculate: n ^X k+1jk = ' t k+1 ; ^X o kjk ; t k ; u ^X (jtk ) ; ; t k+1 t k (26) R R P k+1jk = k+1;k P kjk T k+1;k + Pk+1;k (27) R R where Pk+1;k is a sum inclusive of gravity acceleration error covariance, air-drag acceleration error covariace, solar pressure acceleration error covariance, and thrust acceleration error covariance. R R R R 14

15 5.6 Filter Measurement Update Algorithm The Kalman lter [4] measurement update theorem, and associated algorithm, was a rst order breakthrough for orbit determination. Although its implementation is associated with minor numerical di culties (see Bucy and Joseph [7], page 141 and Chapter 16), its content has enabled, in part, the realization of optimality. Let t k be the time of last measurement y k. Given a new scalar measurement y k+1 at time t k+1 t k, its scalar non-zero measurement error variance R k+1, the state estimate ^X k+1jk, and the state estimate error covariance matrix P k+1jk, calculate: y k+1 = y k+1 y ^Xk+1jk (X) H k+1 = ^X k+1jk ~R k+1 = H k+1 P k+1jk H T k+1 + R k+1 (30) q If jy k+1 j < 3 ~R k+1, Continue (31) K k+1 = P k+1jk H T k+1= ~ R k+1 (32) ^X k+1jk+1 = ^X k+1jk + K k+1 y k+1 (33) P k+1jk+1 = (I K k+1 H k+1 ) P k+1jk (34) Else discard y k+1, acquire y k+2, k + 1! k + 2 (35) Multiple measurements y k+1 = y j k+1, j f1; 2; :::g, may be processed at time t k+1 by repetition of this sequence. Eqs. 31 and 35 de ne the Kalman measurement editor, naturally embedded in the Kalman measurement update algorithm. Notice that ~ R k+1 is a scalar: H k+1 is 1xN, P k+1jk is NxN, and H T k+1 is Nx1. And note that ~ R k+1 is always non-zero because R k+1 is always non-zero. ODTK provides this scalar measurement update as the default, but also provides the capability to perform one simultaneous measurement update for cases when there are multiple measurements at one epoch. 5.7 Filter Measurement Update and Filter Time Update We use the optimal measurement update algorithm due to Kalman and the optimal time update algorithm due to Wright. They are used recursively in a repeating sequential pattern without reprocessing any measurement. The optimal measurement update algorithm always processes, or rejects completely, a new measurement; i.e., it is never iterated on the same measurement. Optimal iteration on the same measurement would require use of an auto-correlation model between the error on each measurement and the same error on the same measurement when reprocessed. 5.8 Measurement Editing Given realistic covariance matrices and the successful completion of lter initalization, the Kalman measurement editor de ned by Eqs. 31 and 35 provides a remarkable capability. But there is one situation where the Kalman editor can fail. 15

16 5.8.1 Kalman Measurement Editor Limitation When the time update function propagates the state error covariance over an extended time interval the error variances can grow very signi cantly due to force modeling errors. Large state error variances are mapped into measurement error variances, and this opens wide the lter editor threshold. If the rst measurement processed is a signi cant outlier and its residual falls within the editor threshold, then the outlier is accepted and incorporated by the measurement update, and state error variances are suddenly reduced. The state estimate is thereby signi cantly corrupted. If the the second measurement and measurements immediately following are good, it is likely that the measurement editor will reject these good measurements. Eventually the state error variances will grow to the point where the lter editor will accept one or more measurements. If these are good measurements, then the lter is autonomously reinitialized and it continues happily along Filter Initialization Problem Typically the state error covariance is unrealistic during the lter initialization time interval because the error covariance matrix associated with the a priori orbit estimate is unknown. Thus the measurement editing thresholds are also unrealistic usually too small. In this case the measurements are thrown out by the Kalman editor, and no measurements can be processed by the lter measurement update function The Dynamic Editor ODTK provides a Dynamic Editor that removes, in part, the Kalman editor limitation. Let N RT (default 10) denote an integer count reject threshold used to trigger the expansion of the measurement editor threshold, and let N AT (default 3) denote an integer count accept threshold used to trigger the contraction of the measurement editor threshold. Integers N RT and N AT are de ned by the user. Let N LO (default 3) denote the ordinary number of sigmas for the Kalman editor threshold, and let N HI (default 100) denote the extrordinary number of sigmas for the Dynamic Editor. N LO is used for the nominal Kalman editor. During operation of the sequential lter if N RT measurements are sequentially rejected, then N HI is used if the user has turned on the Dynamic Editor. If N HI is in use, and if N AT measurements are sequentially accepted, then N HI is switched to N LO. 5.9 Filter Initialization The Dynamic Editor may be used for lter initialization, where N AT is set to a large integer value. After lter initialization, the lter restart capability should be used to modify or eliminate the Dynamic Editor, and the lter should be restarted. Filter divergence is not de ned during lter initialization Filter Divergence Filter divergence is intimately related to both the Kalman measurement editor and the complementary editor. Let N D denote the number of complete measurement sets sequentially rejected by an editor to de ne lter divergence, where the N D integer value is set by the user. A complete measurement set refers to all measurments de ned at the same time tag Kalman Measurement Editor Given completion of lter initialization, and given that the Dynamic Editor is not in use, then lter divergence is de ned after N D complete measurement sets have been sequentially rejected Dynamic Editor Given completion of lter initialization, and given the use of the Dynamic Editor, lter divergence is de ned after (N D + N RT ) complete measurement sets have been sequentially rejected. 16

17 Unmodeled Thrust Maneuvers Any signi cant unmodeled thrust maneuver will cause the lter to immediately diverge. When tracking hostile spacecraft this is a tremendous advantage, for it enables us to determine unknown thrust maneuver schedules for hostile spacecraft. When tracking friendly spacecraft, always tell the lter about upcoming thrust maneuvers. This will enable the lter to optimally ride across the maneuver without a lter restart. 6 Fixed Interval Sequential Smoother (FIS) Inputs to the xed interval sequential smoother are outputs from the sequential lter. These lter outputs must therefore be stored while running the lter, for use in the smoother. The last lter output is the rst smoother input and serves to initialize the smoother. The lter runs forward with time. The smoother runs backwards with time. For this chapter only, we need notation to distinguish state estimates produced by the lter from state estimates produced by the smoother. Both are used in the same equations. Then let ^X and ^P denote state estimate and covariance output by the lter, and let ~ X and ~ P denote state estimate and covariance output by the smoother. The hats ^X and ^P denote lter, and the tildas ~ X and ~ P denote smoother. Let t 0 denote the rst lter time in the xed interval ft 0 ; t L g, and let t L, where t 0 < t L, denote the last lter time in the xed interval ft 0 ; t L g. 6.1 FIS Initialization At the last time t L in the xed interval ft 0 ; t L g, set: ~X LjL = ^X LjL (36) ~P LjL = ^P LjL (37) 6.2 Notation for Smoother Nonlinear State Transition The left-hand side of Eq. 38 provides shorthand notation for propagation of the smoothed state estimate ~X k+1jl backwards in time from t k+1 to t k < t k+1 to get ' ~Xk+1jL k : ' ~Xk+1jL k = ' t k ; X ~ k+1jl ; t k+1 ; u () ; t k t k+1 (38) where k 2 fl 1; L 2; : : : ; 1; 0g. Note that: ~X kjl 6= ' k ~Xk+1jL (39) In order to accomodate the smoother state estimate transition function for orbit substates, the VOP trajectory propagator will run backwards with time. 6.3 FIS Sequential Equations For k 2 fl 1; L 2; : : : ; 1; 0g: Transition Smoothed State Estimate Backwards ' ~Xk+1jL k = ' t k ; X ~ k+1jl ; t k+1 ; u () ; t k t k+1 (40) Incorporate Filter Estimate and Covariance at Time t k ~X kjl = ^X kjk + ^P kjk ^P kjk + ^ 1 ^P RR T 1 h 1 k+1;k k+1;k ^ k+1;k ' ~Xk+1jL k ^X kjk i (41) 17

18 6.3.3 Prepare to Calculate Smoother Covariance Smoother Covariance A k;k+1 = ^P kjk ^T k+1;k ^P 1 k+1jk (42) ~P kjl = ^P kjk + A k;k+1 h ~Pk+1jL ^Pk+1jk i A T k;k+1 (43) The matrix subtraction in Eq. 43 is delicate; i.e., we must guarantee that ~ P kjl has no negative eigenvalues due to numerical round-o. 6.4 Filter-Smoother Consistency Test Considering the FIS algorithm, calculate the N N di erence matrix P kjl between the ltered covariance matrix ^P kjk and the smoothed covariance matrix ~ P kjl for time t k : P kjl = ^P kjk ~ PkjL (44) for each k 2 f0; 1; 2; : : : ; Lg. The di erence matrix P kjl should have no negative eigenvalues. Denote the square root of the i th main diagonal element of the N N di erence matrix P kjl as i kjl. Also calculate the N 1 di erence matrix X kjl between ltered state estimate ^X kjk and smoothed state estimate X ~ kjl for time t k : X kjl = ^X kjk ~ XkjL (45) Denote the i th element of the N 1 di erence matrix X kjl as X kjl i. Now calculate and graph the ratio : R i kjl = X i kjl =i kjl (46) for each i 2 f1; 2; : : : ; Ng and for each k 2 f0; 1; 2; : : : ; Lg. This is the McReynolds lter-smoother test statistic Criterion If for each i 2 f1; 2; : : : ; Ng and for each k 2 f0; 1; 2; : : : ; Lg we have:. 3 (47) R i kjl then McReynolds lter-smoother test criterion is satis ed globally. If for each i 2 f1; 2; : : : ; Ng and for each k 2 f0; 1; 2; : : : ; Lg we have: 3 (48) R i kjl then McReynolds lter-smoother test is failed globally. For each i for which inequality 47 is satis ed McReynolds lter-smoother test is passed for that state estimate element, and for each i for which inequality 48 is satis ed McReynolds lter-smoother test is failed for that state estimate element. 7 Variable-Lag Smoother A lter that runs forward with time, followed by an extended FIS that runs backward with time, cannot be conveniently used for near-real-time estimation because all measurements of interest across the xed-interval must be completely ltered and smoothed before a smoothed state is estimated. This demonstrate the need for a smoothing lag that is time-variable. The extension of linear estimators to non-linear problems can be achieved with two very di erent techniques. One technique propagates variations of the state estimate with a linear transition matrix function. However, the propagation of the state estimate directly with a non-linear transition function ' cannot always be achieved with sequential smoothers. The two smoothing algorithms now described for use in ODTK s variable-lag smoother employ the linear transition matrix function. 18

19 8 Carlton-Rauch Fixed-Epoch Smoother (CR-FES) 8.1 FES Initialization Let t FE denote a xed epoch, coincident with time centroid of an event of interest known a priori, such as an impulsive spacecraft maneuver. Let ^X FE and P FE denote ltered state estimate and covariance at t FE. Meditch s presentation of the Carlton-Rauch FES denotes t FE as coincident with time-tag of some measurement processed; i.e., t FE = t k for ^X FE = ^X kjk and P FE = P kjk, but this is not necessary 3. It may be necessary that ^X FE = ^X kjk 1 and P FE = P kjk 1 for propagated state estimate ^X kjk 1 and covariance P kjk 1. However, we continue with Meditch s notation to be consistent with that presentation. Let ^X kjj 1 denote an FES state estimate with xed epoch t k where j = k+1, where the last measurement processed by the lter has time-tag t k or t k 1, and where t k t j 1 or t k 1 t j 1 respectively. With the lter at time t FE = t k, initialize the FES by storing objects associated with, or calculated by, the lter. Store t k and ^X kjj 1 = ^X FE (49) P kjj 1 = P FE (50) ^X j 1jj 1 = ^X FE (51) P j 1jj 1 = P FE (52) B j 1 = I (53) 8.2 Measurement at t j = t k+1 For this section j = k + 1, where t k is the xed epoch and t j > t k is the time-tag for a new measurement y j = y k Filter The lter calculates the propagated state estimate ^X k+1jk = ^X jjj 1, propagated covariance P k+1jk = P jjj 1, ltered state estimate ^X k+1jk+1 = ^X jjj, ltered covariance P k+1jk+1 = P jjj, and transition matrix k+1;k = j;j 1. Store ^X jjj 1, P jjj 1, ^Xjjj, P jjj, and j;j 1 for use in the FES. For the rst value of B j, following FES initialization, set where B j = B j 1 A j 1 (54) A j 1 = P j 1jj 1 T j;j 1P 1 jjj 1 (55) CR-FES FES calculations refer to the xed epoch t k, and to lter measurement time-tags t j t k. ^X kjj = ^X kjj 1 + B j h ^Xjjj ^Xjjj 1 i (56) P kjj = P kjj 1 + B j Pjjj P jjj 1 B T j (57) 3 If one processes a pseudo-measurement with time-tag t F E that has zero measurement-state partial derivatives, then Meditch s notation is maintained and no harm is done to the VLS. 19

20 If the column matrix ^X kjj has n elements, then P kjj, P kjj 1, P jjj 1, P jjj, B j, and T j;j 1 are n n matrices. Covariance matrices P kjj, P kjj 1, P jjj 1, and P jjj are symmetric and are free of negative eigenvalues. Zero eigenvalues in P jjj 1 are not acceptable because it must be inverted. The implementation must guarantee that symmetric matrices are numerically symmetric, that all covariance matrices are numerically free of negative eigenvalues, and that P jjj 1 is free of zero eigenvalues Filter After FES execution and recording of FES results, the FES recursion is performed by the lter in preparation for the next measurement. ^X kjj 1 = ^X kjj (58) P kjj 1 = P kjj (59) 8.3 Measurements at t j = t k+1 ; t k+2 ; : : : B j 1 = B j (60) In the section Measurement at t j = t k+1 above, replace t k+1 with t k+2 for the measurement y j = y k+2 at time t j = t k+2. When t j = t k+h, replace t k+1 with t k+h for the measurement y j = y k+h at time t j = t k+h. 9 Frazer Fixed-Epoch Smoother (F-FES) 9.1 FES Initialization Nomenclature follows that of the CR-FES. With the lter at time t FE = t k, initialize the FES by storing objects associated with, or calculated by, the lter: t k, ^Xkjj 1 = ^X FE, P kjj 1 = P FE, and W j 1 = P FE. 9.2 Measurement at t j = t k+1 For this section j = k + 1, where t k is the xed epoch and t j > t k is the time-tag for a new measurement y j = y k Filter The lter calculates the propagated state estimate ^X k+1jk = ^X jjj 1, propagated covariance P k+1jk = P jjj 1, ltered state estimate ^X k+1jk+1 = ^X jjj, ltered covariance P k+1jk+1 = P jjj, transition matrix k+1;k = j;j 1, measurement-state jacobian matrix H k+1 = H j, measurement covariance matrix R k+1 = R j, and measurement residual y k+1;k = y j;j 1 at time-tag t k+1 = t j for the new measurement y k+1 = y j. Store ^X jjj 1, P jjj 1, X jjj, P jjj, j;j 1, H j, R j, and y j;j 1 for use in the FES F-FES The following algorithm was constructed by Frazer ([5] Corollary 6.1 page 232). Fixed epoch smoother calculations refer to the xed epoch t k, and to lter measurement time-tags t j t k. S j = H T j R 1 j H j (61) W j = W j 1 T j;j 1 I Sj P jjj (62) ^X kjj = ^X kjj 1 + W j H T j R 1 j y j;j 1 (63) 20

21 P kjj = P kjj 1 W j Sj P jjj 1 S j + S j W T j (64) If the column matrix ^X kjj has n elements, then P kjj, P kjj 1, P jjj 1, P jjj, T j;j 1, S j, and W j are n n matrices. Covariance matrices P kjj, P kjj 1, P jjj 1, and P jjj are symmetric matrices with positive or zero eigenvalues. Zero covariance matrix eigenvalues are acceptable because no state-sized covariance matrix inverse is required for Frazer form of the FES. S j is seen to be symmetric by inspection of it s de ning Equation 61. The implementation must guarantee that symmetric matrices are numerically symmetric, and that covariance matrices are numerically free of negative eigenvalues. W j 1 is initialized as a symmetric covariance matrix, but W j and subsequent W j 1 matrices are not symmetric due to the factor T j;j 1 in the recursive Equation 62. F-FES calculations require products of matrices with extreme di erences in order of magnitude. For example, the calculation of W j according to Equation 62 requires evaluation of the product S j P jjj that is subtracted from a matrix of order unity. The eigenvalues of S j are very large due to the small values of R j, and the eigenvalues of P jjj are very small, all non-negative. The product S j P jjj is of order unity, but some signi cance is lost in double precision calculations. It may thus be advisable to premultiply S j by a small positive scalar and premultiply P jjj by it s inverse 1 for calculation of the product S j P jjj = (S j ) 1 P jjj, where (Sj ) and 1 P jjj are both of order unity Filter After FES execution and recording of FES results, the FES recursion is performed by the lter in preparation for the next measurement. ^X kjj 1 = ^X kjj (65) P kjj 1 = P kjj (66) 9.3 Measurements at t j = t k+1 ; t k+2 ; : : : W j 1 = W j (67) In the section Measurement at t j = t k+1 above, replace t k+1 with t k+2 for the measurement y j = y k+2 at time t j = t k+2. When t j = t k+h, replace t k+1 with t k+h for the measurement y j = y k+h at time t j = t k+h. 10 Least Squares Orbit determination using a batch of measurements from a xed time interval could use the classical least squares normal equation: A T A X = A T b (68) where A is the given triple product of an m m root diagonal matrix W 1=2 of measurement weights, an m n matrix of partial derivatives, and an n n state error transition matrix. Matrix A has rank n with m n. The given m 1 matrix b is the product of matrix W 1=2 with the m 1 matrix y of measurement residuals. X is the unknown n 1 matrix to be calculated. Conventional solution: X = A T A 1 A T b (69) But dense range measurement values may have variations only in the last few signi cant decimals. Given double precision calculation with 15 + decimal mantissas, generally half of the 15 + decimal signi cance is lost due to the squaring operation in the calculation of A T A. This signi cance is regained by solving instead, the equation: 21

22 for X. This seems trivial when m = n : A X = b (70) X = A 1 b (71) For m > n, the solution of Eq. 70 with greatest numerical stability requires the use of orthogonal Householder transformations (Lawson & Hanson [40], page 121) for triangularization of matrix A. It is important to note that Eqs. 68 and 70 are theoretically equivalent in the sense that each implies the other. Our problem is to calculate an acceptable solution of Eq. 70 when m > n. Triangularization T A of matrix A is obtained with the calculation of an m m orthogonal matrix T such that the upper n n matrix of the m n matrix T A is upper triangular, and the lower (m n) n matrix of the m n matrix T A is zero. Multiply Eq. 70 through by matrix T to get: T A X = T b (72) This is easily solved for X with back substitutions because T A is upper triangular. Part II Computer Science 11 Software Architecture The ODTK Application software design consists of a multi-tiered, layered architecture. Key design goals are to: Reuse and share astrodynamics functionality with the STK code libraries to achieve consistent mathematical modeling between both the STK and ODTK products; Maintain separation between the Graphical User Interface (GUI) and engine software to minimize complexity and improve portability; Provide enhanced user interface controls and a consistent look and feel with other AGI products. The gure and sections below describe the ODTK architecture in more detail. 12 AGI Technology Platform (ATP) The ATP provides an expandable application and component framework for the ODTK software. ODTK makes use of the following ATP Technologies: Application "personality" Object browser - for viewing the OD con guration objects Property editor - for viewing and editing the OD object attributes Enhanced GUI controls, Attributes: an advanced symbolic access functionality that provides a consistent and logical objectbased scheme for getting and setting internal data. Scripting automation support Reporting and graphing 22